We construct via generators and relations a generalized Weil representation for the split orthogonal group $\ot_q(2n,2n)$ over a finite field of $q$ elements. Besides, we give an initial decomposition of the representation found. We also show that the constructed representation is equal to the restriction of the Weil representation to $\ot_q(2n,2n)$ for the reductive dual pair $(\Sp_2(\mathbb{ F}_q),\ot_q(2n,2n))$ and that the initial decomposition is the same as the decomposition with respect to the action of $\Sp_2(\mathbb{ F}_q)$.
Categories: math.RT